Special Relativity | Part IB, 2001

What is Einstein's principle of relativity?

Show that a spherical shell expanding at the speed of light, x2=c2t2\mathbf{x}^{2}=c^{2} t^{2}, in one coordinate system (t,x)(t, \mathbf{x}), is still spherical in a second coordinate system (t,x)\left(t^{\prime}, \mathbf{x}^{\prime}\right) defined by

ct=γ(ctucx)x=γ(xut)y=yz=z\begin{aligned} c t^{\prime} &=\gamma\left(c t-\frac{u}{c} x\right) \\ x^{\prime} &=\gamma(x-u t) \\ y^{\prime} &=y \\ z^{\prime} &=z \end{aligned}

where γ=(1u2/c2)12\gamma=\left(1-u^{2} / c^{2}\right)^{-\frac{1}{2}}. Why do these equations define a Lorentz transformation?

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